Ostrogradsky instability

In applied mathematics, the Ostrogradsky instability is a feature of some solutions of theories having equations of motion with more than two time derivatives (higher-derivative theories). It is suggested by a theorem of Mikhail Ostrogradsky in classical mechanics according to which a non-degenerate Lagrangian dependent on time derivatives higher than the first corresponds to a Hamiltonian unbounded from below. As usual, the Hamiltonian is associated with the Lagrangian via a Legendre transform. The Ostrogradsky instability has been proposed as an explanation as to why no differential equations of higher order than two appear to describe physical phenomena. However, Ostrogradsky's theorem does not imply that all solutions of higher-derivative theories are unstable as many counterexamples are known.